×

The space of \(p\)-summable sequences and its natural \(n\)-norm. (English) Zbl 1002.46007

Let \(n\) be a nonnegative integer. In this note, the author studies the space \(\ell^p\), \(1 \leq p \leq \infty,\) equipped the natural \(n\)-norm in the sense of A. Misiak [Math. Nachr. 140, 299-319 (1989; Zbl 0673.46012)]. The concept of an \(n\)-norm is a generalization of the concept of a \(2\)-norm developed by S. Gähler [Math. Nach. 28, 1-43 (1964; Zbl 0142.39803)]. It is shown in the paper that \(\ell^p\), \(1 \leq p \leq \infty,\) is complete with respect to the \(n\)-norm. The author also proves a fixed point theorem for \(\ell^p\) as an \(n\)-normed space.

MSC:

46A70 Saks spaces and their duals (strict topologies, mixed topologies, two-norm spaces, co-Saks spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
46B99 Normed linear spaces and Banach spaces; Banach lattices
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Naidu, Indian J. Pure Appl. Math. 17 pp 974– (1986)
[2] Siddiqi, Indian J. Math. 31 pp 31– (1989)
[3] Malčeski, Mat. Bilten 21 pp 103– (1997)
[4] Ganguly, J. Indian Acad. Math. 4 pp 80– (1982)
[5] DOI: 10.1002/mana.19690410103 · Zbl 0182.56601 · doi:10.1002/mana.19690410103
[6] DOI: 10.1002/mana.19690400405 · Zbl 0182.56501 · doi:10.1002/mana.19690400405
[7] DOI: 10.1002/mana.19690400114 · Zbl 0182.56404 · doi:10.1002/mana.19690400114
[8] DOI: 10.1002/mana.19640280102 · Zbl 0142.39803 · doi:10.1002/mana.19640280102
[9] DOI: 10.1002/mana.19630260109 · Zbl 0117.16003 · doi:10.1002/mana.19630260109
[10] Diminnie, Demonstratio Math. 10 pp 169– (1977)
[11] Diminnie, Demonstratio Math. 6 pp 525– (1973)
[12] Browna, Elements of functional analysis (1970)
[13] Malčeski, Mat. Bilten 21 pp 81– (1997)
[14] Lai, Bull. Austral. Math. Soc. 18 pp 137– (1978)
[15] Kim, Demonstratio Math. 29 pp 739– (1996)
[16] DOI: 10.1155/S0161171293000547 · Zbl 0829.47048 · doi:10.1155/S0161171293000547
[17] Jain, Far East J. Math. Sci. 3 pp 51– (1995)
[18] Greub, Linear algebra (1975) · doi:10.1007/978-1-4684-9446-4
[19] Tewari, Indian J. Math. 25 pp 161– (1983)
[20] Suyalatu, Natur. Sci. J. Harbin Normal Univ. 6 pp 20– (1990)
[21] DOI: 10.1002/mana.19891400121 · Zbl 0673.46012 · doi:10.1002/mana.19891400121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.